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<h2 title="Class AdamsMoultonIntegrator" class="title">Class AdamsMoultonIntegrator</h2>
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<dd><a href="../../../../../../org/apache/commons/math3/ode/FirstOrderIntegrator.html" title="interface in org.apache.commons.math3.ode">FirstOrderIntegrator</a>, <a href="../../../../../../org/apache/commons/math3/ode/ODEIntegrator.html" title="interface in org.apache.commons.math3.ode">ODEIntegrator</a></dd>
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<pre>public class <span class="strong">AdamsMoultonIntegrator</span>
extends <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff">AdamsIntegrator</a></pre>
<div class="block">This class implements implicit Adams-Moulton integrators for Ordinary
Differential Equations.
<p>Adams-Moulton methods (in fact due to Adams alone) are implicit
multistep ODE solvers. This implementation is a variation of the classical
one: it uses adaptive stepsize to implement error control, whereas
classical implementations are fixed step size. The value of state vector
at step n+1 is a simple combination of the value at step n and of the
derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to
compute y<sub>n+1</sub>, another method must be used to compute a first
estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute
a final estimate of y<sub>n+1</sub> using the following formulas. Depending
on the number k of previous steps one wants to use for computing the next
value, different formulas are available for the final estimate:</p>
<ul>
<li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li>
<li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li>
<li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (5y'<sub>n+1</sub>+8y'<sub>n</sub>-y'<sub>n-1</sub>)/12</li>
<li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (9y'<sub>n+1</sub>+19y'<sub>n</sub>-5y'<sub>n-1</sub>+y'<sub>n-2</sub>)/24</li>
<li>...</li>
</ul>
<p>A k-steps Adams-Moulton method is of order k+1.</p>
<h3>Implementation details</h3>
<p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
<pre>
s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
...
s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
</pre></p>
<p>The definitions above use the classical representation with several previous first
derivatives. Lets define
<pre>
q<sub>n</sub> = [ s<sub>1</sub>(n-1) s<sub>1</sub>(n-2) ... s<sub>1</sub>(n-(k-1)) ]<sup>T</sup>
</pre>
(we omit the k index in the notation for clarity). With these definitions,
Adams-Moulton methods can be written:
<ul>
<li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li>
<li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 1/2 s<sub>1</sub>(n+1) + [ 1/2 ] q<sub>n+1</sub></li>
<li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 5/12 s<sub>1</sub>(n+1) + [ 8/12 -1/12 ] q<sub>n+1</sub></li>
<li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 9/24 s<sub>1</sub>(n+1) + [ 19/24 -5/24 1/24 ] q<sub>n+1</sub></li>
<li>...</li>
</ul></p>
<p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with
higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
and r<sub>n</sub>) where r<sub>n</sub> is defined as:
<pre>
r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
</pre>
(here again we omit the k index in the notation for clarity)
</p>
<p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
for degree k polynomials.
<pre>
s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + ∑<sub>j>0</sub> (j+1) (-i)<sup>j</sup> s<sub>j+1</sub>(n)
</pre>
The previous formula can be used with several values for i to compute the transform between
classical representation and Nordsieck vector. The transform between r<sub>n</sub>
and q<sub>n</sub> resulting from the Taylor series formulas above is:
<pre>
q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
</pre>
where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built
with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being
the column number starting from 1:
<pre>
[ -2 3 -4 5 ... ]
[ -4 12 -32 80 ... ]
P = [ -6 27 -108 405 ... ]
[ -8 48 -256 1280 ... ]
[ ... ]
</pre></p>
<p>Using the Nordsieck vector has several advantages:
<ul>
<li>it greatly simplifies step interpolation as the interpolator mainly applies
Taylor series formulas,</li>
<li>it simplifies step changes that occur when discrete events that truncate
the step are triggered,</li>
<li>it allows to extend the methods in order to support adaptive stepsize.</li>
</ul></p>
<p>The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step
n as follows:
<ul>
<li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
<li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li>
<li>R<sub>n+1</sub> = (s<sub>1</sub>(n) - S<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub></li>
</ul>
where A is a rows shifting matrix (the lower left part is an identity matrix):
<pre>
[ 0 0 ... 0 0 | 0 ]
[ ---------------+---]
[ 1 0 ... 0 0 | 0 ]
A = [ 0 1 ... 0 0 | 0 ]
[ ... | 0 ]
[ 0 0 ... 1 0 | 0 ]
[ 0 0 ... 0 1 | 0 ]
</pre>
From this predicted vector, the corrected vector is computed as follows:
<ul>
<li>y<sub>n+1</sub> = y<sub>n</sub> + S<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... ±1 ] r<sub>n+1</sub></li>
<li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
<li>r<sub>n+1</sub> = R<sub>n+1</sub> + (s<sub>1</sub>(n+1) - S<sub>1</sub>(n+1)) P<sup>-1</sup> u</li>
</ul>
where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the
predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub>
represent the corrected states.</p>
<p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
they only depend on k and therefore are precomputed once for all.</p></div>
<dl><dt><span class="strong">Since:</span></dt>
<dd>2.0</dd></dl>
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<code><a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.NordsieckTransformer.html" title="interface in org.apache.commons.math3.ode">MultistepIntegrator.NordsieckTransformer</a></code></li>
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<code><a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html#nordsieck">nordsieck</a>, <a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html#scaled">scaled</a></code></li>
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<code><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#mainSetDimension">mainSetDimension</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#scalAbsoluteTolerance">scalAbsoluteTolerance</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#scalRelativeTolerance">scalRelativeTolerance</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#vecAbsoluteTolerance">vecAbsoluteTolerance</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#vecRelativeTolerance">vecRelativeTolerance</a></code></li>
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<code><a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#isLastStep">isLastStep</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#resetOccurred">resetOccurred</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#stepHandlers">stepHandlers</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#stepSize">stepSize</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#stepStart">stepStart</a></code></li>
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<td class="colOne"><code><strong><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsMoultonIntegrator.html#AdamsMoultonIntegrator(int,%20double,%20double,%20double[],%20double[])">AdamsMoultonIntegrator</a></strong>(int nSteps,
double minStep,
double maxStep,
double[] vecAbsoluteTolerance,
double[] vecRelativeTolerance)</code>
<div class="block">Build an Adams-Moulton integrator with the given order and error control parameters.</div>
</td>
</tr>
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<td class="colOne"><code><strong><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsMoultonIntegrator.html#AdamsMoultonIntegrator(int,%20double,%20double,%20double,%20double)">AdamsMoultonIntegrator</a></strong>(int nSteps,
double minStep,
double maxStep,
double scalAbsoluteTolerance,
double scalRelativeTolerance)</code>
<div class="block">Build an Adams-Moulton integrator with the given order and error control parameters.</div>
</td>
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<td class="colFirst"><code>void</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsMoultonIntegrator.html#integrate(org.apache.commons.math3.ode.ExpandableStatefulODE,%20double)">integrate</a></strong>(<a href="../../../../../../org/apache/commons/math3/ode/ExpandableStatefulODE.html" title="class in org.apache.commons.math3.ode">ExpandableStatefulODE</a> equations,
double t)</code>
<div class="block">Integrate a set of differential equations up to the given time.</div>
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<h3>Methods inherited from class org.apache.commons.math3.ode.nonstiff.<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff">AdamsIntegrator</a></h3>
<code><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsIntegrator.html#initializeHighOrderDerivatives(double,%20double[],%20double[][],%20double[][])">initializeHighOrderDerivatives</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsIntegrator.html#updateHighOrderDerivativesPhase1(org.apache.commons.math3.linear.Array2DRowRealMatrix)">updateHighOrderDerivativesPhase1</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsIntegrator.html#updateHighOrderDerivativesPhase2(double[],%20double[],%20org.apache.commons.math3.linear.Array2DRowRealMatrix)">updateHighOrderDerivativesPhase2</a></code></li>
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<h3>Methods inherited from class org.apache.commons.math3.ode.<a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html" title="class in org.apache.commons.math3.ode">MultistepIntegrator</a></h3>
<code><a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html#computeStepGrowShrinkFactor(double)">computeStepGrowShrinkFactor</a>, <a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html#getMaxGrowth()">getMaxGrowth</a>, <a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html#getMinReduction()">getMinReduction</a>, <a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html#getNSteps()">getNSteps</a>, <a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html#getSafety()">getSafety</a>, <a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html#getStarterIntegrator()">getStarterIntegrator</a>, <a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html#setMaxGrowth(double)">setMaxGrowth</a>, <a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html#setMinReduction(double)">setMinReduction</a>, <a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html#setSafety(double)">setSafety</a>, <a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html#setStarterIntegrator(org.apache.commons.math3.ode.FirstOrderIntegrator)">setStarterIntegrator</a>, <a href="../../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html#start(double,%20double[],%20double)">start</a></code></li>
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<h3>Methods inherited from class org.apache.commons.math3.ode.nonstiff.<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff">AdaptiveStepsizeIntegrator</a></h3>
<code><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#filterStep(double,%20boolean,%20boolean)">filterStep</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#getCurrentStepStart()">getCurrentStepStart</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#getMaxStep()">getMaxStep</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#getMinStep()">getMinStep</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#initializeStep(boolean,%20int,%20double[],%20double,%20double[],%20double[],%20double[],%20double[])">initializeStep</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#resetInternalState()">resetInternalState</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#sanityChecks(org.apache.commons.math3.ode.ExpandableStatefulODE,%20double)">sanityChecks</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#setInitialStepSize(double)">setInitialStepSize</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#setStepSizeControl(double,%20double,%20double[],%20double[])">setStepSizeControl</a>, <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html#setStepSizeControl(double,%20double,%20double,%20double)">setStepSizeControl</a></code></li>
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<h3>Methods inherited from class org.apache.commons.math3.ode.<a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html" title="class in org.apache.commons.math3.ode">AbstractIntegrator</a></h3>
<code><a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#acceptStep(org.apache.commons.math3.ode.sampling.AbstractStepInterpolator,%20double[],%20double[],%20double)">acceptStep</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#addEventHandler(org.apache.commons.math3.ode.events.EventHandler,%20double,%20double,%20int)">addEventHandler</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#addEventHandler(org.apache.commons.math3.ode.events.EventHandler,%20double,%20double,%20int,%20org.apache.commons.math3.analysis.solvers.UnivariateSolver)">addEventHandler</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#addStepHandler(org.apache.commons.math3.ode.sampling.StepHandler)">addStepHandler</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#clearEventHandlers()">clearEventHandlers</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#clearStepHandlers()">clearStepHandlers</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#computeDerivatives(double,%20double[],%20double[])">computeDerivatives</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#getCounter()">getCounter</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#getCurrentSignedStepsize()">getCurrentSignedStepsize</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#getEvaluations()">getEvaluations</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#getEvaluationsCounter()">getEvaluationsCounter</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#getEventHandlers()">getEventHandlers</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#getExpandable()">getExpandable</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#getMaxEvaluations()">getMaxEvaluations</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#getName()">getName</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#getStepHandlers()">getStepHandlers</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#initIntegration(double,%20double[],%20double)">initIntegration</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#integrate(org.apache.commons.math3.ode.FirstOrderDifferentialEquations,%20double,%20double[],%20double,%20double[])">integrate</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#setEquations(org.apache.commons.math3.ode.ExpandableStatefulODE)">setEquations</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#setMaxEvaluations(int)">setMaxEvaluations</a>, <a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#setStateInitialized(boolean)">setStateInitialized</a></code></li>
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<h3>Methods inherited from class java.lang.<a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></h3>
<code><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#clone()" title="class or interface in java.lang">clone</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#equals(java.lang.Object)" title="class or interface in java.lang">equals</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#finalize()" title="class or interface in java.lang">finalize</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#getClass()" title="class or interface in java.lang">getClass</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#hashCode()" title="class or interface in java.lang">hashCode</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notify()" title="class or interface in java.lang">notify</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notifyAll()" title="class or interface in java.lang">notifyAll</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#toString()" title="class or interface in java.lang">toString</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait()" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long)" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long,%20int)" title="class or interface in java.lang">wait</a></code></li>
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<a name="AdamsMoultonIntegrator(int, double, double, double, double)">
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<h4>AdamsMoultonIntegrator</h4>
<pre>public AdamsMoultonIntegrator(int nSteps,
double minStep,
double maxStep,
double scalAbsoluteTolerance,
double scalRelativeTolerance)
throws <a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooSmallException.html" title="class in org.apache.commons.math3.exception">NumberIsTooSmallException</a></pre>
<div class="block">Build an Adams-Moulton integrator with the given order and error control parameters.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>nSteps</code> - number of steps of the method excluding the one being computed</dd><dd><code>minStep</code> - minimal step (sign is irrelevant, regardless of
integration direction, forward or backward), the last step can
be smaller than this</dd><dd><code>maxStep</code> - maximal step (sign is irrelevant, regardless of
integration direction, forward or backward), the last step can
be smaller than this</dd><dd><code>scalAbsoluteTolerance</code> - allowed absolute error</dd><dd><code>scalRelativeTolerance</code> - allowed relative error</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooSmallException.html" title="class in org.apache.commons.math3.exception">NumberIsTooSmallException</a></code> - if order is 1 or less</dd></dl>
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<h4>AdamsMoultonIntegrator</h4>
<pre>public AdamsMoultonIntegrator(int nSteps,
double minStep,
double maxStep,
double[] vecAbsoluteTolerance,
double[] vecRelativeTolerance)
throws <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/IllegalArgumentException.html?is-external=true" title="class or interface in java.lang">IllegalArgumentException</a></pre>
<div class="block">Build an Adams-Moulton integrator with the given order and error control parameters.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>nSteps</code> - number of steps of the method excluding the one being computed</dd><dd><code>minStep</code> - minimal step (sign is irrelevant, regardless of
integration direction, forward or backward), the last step can
be smaller than this</dd><dd><code>maxStep</code> - maximal step (sign is irrelevant, regardless of
integration direction, forward or backward), the last step can
be smaller than this</dd><dd><code>vecAbsoluteTolerance</code> - allowed absolute error</dd><dd><code>vecRelativeTolerance</code> - allowed relative error</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/IllegalArgumentException.html?is-external=true" title="class or interface in java.lang">IllegalArgumentException</a></code> - if order is 1 or less</dd></dl>
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<a name="integrate(org.apache.commons.math3.ode.ExpandableStatefulODE, double)">
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<h4>integrate</h4>
<pre>public void integrate(<a href="../../../../../../org/apache/commons/math3/ode/ExpandableStatefulODE.html" title="class in org.apache.commons.math3.ode">ExpandableStatefulODE</a> equations,
double t)
throws <a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooSmallException.html" title="class in org.apache.commons.math3.exception">NumberIsTooSmallException</a>,
<a href="../../../../../../org/apache/commons/math3/exception/DimensionMismatchException.html" title="class in org.apache.commons.math3.exception">DimensionMismatchException</a>,
<a href="../../../../../../org/apache/commons/math3/exception/MaxCountExceededException.html" title="class in org.apache.commons.math3.exception">MaxCountExceededException</a>,
<a href="../../../../../../org/apache/commons/math3/exception/NoBracketingException.html" title="class in org.apache.commons.math3.exception">NoBracketingException</a></pre>
<div class="block">Integrate a set of differential equations up to the given time.
<p>This method solves an Initial Value Problem (IVP).</p>
<p>The set of differential equations is composed of a main set, which
can be extended by some sets of secondary equations. The set of
equations must be already set up with initial time and partial states.
At integration completion, the final time and partial states will be
available in the same object.</p>
<p>Since this method stores some internal state variables made
available in its public interface during integration (<a href="../../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html#getCurrentSignedStepsize()"><code>AbstractIntegrator.getCurrentSignedStepsize()</code></a>), it is <em>not</em> thread-safe.</p></div>
<dl>
<dt><strong>Specified by:</strong></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsIntegrator.html#integrate(org.apache.commons.math3.ode.ExpandableStatefulODE,%20double)">integrate</a></code> in class <code><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff">AdamsIntegrator</a></code></dd>
<dt><span class="strong">Parameters:</span></dt><dd><code>equations</code> - complete set of differential equations to integrate</dd><dd><code>t</code> - target time for the integration
(can be set to a value smaller than <code>t0</code> for backward integration)</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooSmallException.html" title="class in org.apache.commons.math3.exception">NumberIsTooSmallException</a></code> - if integration step is too small</dd>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/DimensionMismatchException.html" title="class in org.apache.commons.math3.exception">DimensionMismatchException</a></code> - if the dimension of the complete state does not
match the complete equations sets dimension</dd>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/MaxCountExceededException.html" title="class in org.apache.commons.math3.exception">MaxCountExceededException</a></code> - if the number of functions evaluations is exceeded</dd>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/NoBracketingException.html" title="class in org.apache.commons.math3.exception">NoBracketingException</a></code> - if the location of an event cannot be bracketed</dd></dl>
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